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Homework Help: Characteristic polynomial help

  1. Nov 1, 2005 #1

    Note: There is exactly one real zero of the characteristic polynomial and it
    has multiplicity 3 (it is a positive integer!). The other zeros are complex
    and they have multiplicity 2.

    Sadly I missed this lecture day, and am unsure of where to start. Any diffeq demi-gods out there?
  2. jcsd
  3. Nov 1, 2005 #2


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    Homework Helper

    For an equation of order n, if a root (say r1) has a multiplicity s (s =< n), where x is the independent variable

    [tex] e^{r_{1}x}, xe^{r_{1}x}, x^{2} e^{r_{1}x}, ..., x^{s-1} e^{r_{1}x} [/tex]

    For complex roots, let's say [itex] a+bi [/itex] is repeated s times, then the complex conjugate [itex] a-bi [/itex] is also repeated s times, therefore the solutions for real valued functions, where x is the independent variable:

    [tex] e^{ax} \cos{bx}, e^{ax} \sin{bx}, xe^{ax} \cos{bx}, xe^{ax} \sin{bx},..., x^{s-1} e^{ax} \cos{bx}, x^{s-1} e^{ax} \sin{bx} [/tex]
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